In this model, the whole hyperbolic plane is shown inside of a disk that can comfortably fit on your screen. And thus, we have to resort to using tricks like the Poincaré disk model, which is what HyperRogue uses. Unfortunately, hyperbolic plane cannot be faithfully represented in our world since its curvature is smaller than zero. On hyperbolic plane, the sum of angles in a triangle is smaller than 180 degrees, and once again, there's direct correlation between the "defect" (180 minus sum of angles) and area. In a sense, it's curvature is actually smaller than of our common Euclidean space, which has curvature of zero. Hyperbolic plane, on the other hand, has NEGATIVE curvature. Not only that, but there will be a direct correlation between its "excess" (sum of angles minus 180) and its area - bigger triangles have bigger excess. This means that a triangle drawn on a surface of a sphere will have sum of its angles greater than 180 degrees. A sphere is an example of a surface with POSITIVE curvature. We all know one example of a non-Euclidean geometry the spherical geometry that governs distances and angles on the surface of Earth, at least if the distances involved are large enough. HyperRogue is a game played in one of these geometries - a hyperbolic geometry - and before you can hope to understand it, you must first understand some basic facts about hyperbolic geometry and the way it's represented in-game. But in the process of finding out that the Fifth Postulate is, indeed, essential for Euclidean geometry, a whole new world of non-Euclidean geometries was discovered. "If you have a straight line and a point outside that line, you can construct exactly one parallel line through that point."Ĭompared to other Euclid's postulates (like "All right angles are equal"), this seemed unneccesarily complex. "Sum of the angles in a triangle is 180 degrees." This postulate has been stated in many equivalent forms, with two of the best-known are: For millenia, best mathematical minds were puzzled by one problem: so-called "Euclid's Fifth Postulate", one of the cornerstones of geometry.
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